Quantitation in analytical chemistry is usually achieved using external calibration. In the presence of matrix interferences, however, the method of internal calibration is used to reduce or eliminate the various sources of errors. Two strategies are available to achieve this: method of standard additions and method of internal standard. The former rests on building the calibration curve within the sample. With all its benefits, standard additions rely on signal intensity measurements and as such, are prone to instrumental drifts and variations in analyte recovery during extraction or separation. To reduce the measurement uncertainty due to instrumental drifts and analyte recoveries, ratio methods are used where all signals are normalized to the internal standard. Isotope dilution is a combination of these two methods utilizing an isotopically labeled internal standard with known amounts. One other difference, however, remains—internal calibration methods provide with the amount of analyte at the time of spike addition whereas external calibration methods yield the amount of analyte at the time of analysis. Therefore, if one is interested in the amount of analyte at the time of analysis using isotope dilution, it must be deduced mathematically or additional spiking experiments need to be carried out as in post-column spiking [Heumann 1998; Meija 2008a]. While majority of analysis are concerned with the amount of analyte at the time of sampling, it is useful to determine the amount of analyte at the time of analysis to judge the quality of analytical methods.
Biologists and sociologists almost always face the question of how to estimate the size of a population known to exist without being able to sample the population entirely. Further, it is rather challenging to account for changes in population size during the analysis. In biology this occurs as birth or death of animals and in chemistry as the loss or the formation of the analyte during the sample analysis. Addition of not just one but multiple spikes of known amounts efficiently solves the problem of quantifying inter-converting analytes [Kingston 1995; Kingston 1998]. In essence, when substances B and C, for example, are known to produce analyte A after addition of isotopically enriched A to the sample, accurate initial amount of substance A can be obtained only when known amounts of enriched substances B and C are also added (hence, multiple-spiking isotope dilution) and all three substances A, B, C can be then measured. The measurand in isotope dilution is the amount of substance (at the time of spiking) and the measured quantity is the isotope pattern of analyte(s), more specifically, isotope ratios. Isotope dilution has been practiced for a long time, initially using radioactive isotopes of lead as) spikes (tracers).
Multiple spiking isotope dilution methods are not uncommon in analytical chemistry, yet the uptake of this advanced calibration approach is slow due to the complexity of the mathematical equations. Currently, several mathematical strategies exist to address simultaneous species formation and degradation using multiple spiking isotope dilution mass spectrometry. Numerous examples of published literature reveal equations that fill entire pages for two or three component systems and the reader is still left without the explicit expressions for the estimates of the measurand [Ruiz Encinar 2002; Point 2007; Monperrus 2008; Van 2008; Rodríguez-González 2004; Tirez 2003]. Such complexity is unwarranted and impedes development of ingenious applications of isotope dilution.
While many of these strategies have been compared numerically, conceptual comparison of the underlying principles is lacking. Due to the recent interest in using the species inter-conversion factors, mainly to study the quality of analytical methods, a review of the mathematical logic and inconsistencies of the existing double or triple spiking isotope dilution models is useful before providing a new model for multiple spiking isotope dilution mass spectrometry. Further, it is useful to provide systematic concepts to clarify the species inter-conversion coefficient definitions currently lacking in elemental speciation.
The application of species-specific isotope dilution has a long history, dating back to as far as 1934, yet all the quantitation applications of this technique traditionally rested entirely on a single salient feature of this technique—the ability to correct for species degradation during the sample analysis. It was not until the mid-1990's when the opposite process, analyte formation during the analysis, received serious attention. Kingston et al. showed first in 1994 that, while conventional isotope dilution methods do correct for species degradation, they are ineffective against the bias introduced from the formation of analyte during the analysis. The potential for the formation of analyte during the analysis is now a widely acknowledged in analytical chemistry. It is observed, for example, during the analysis of Cr(VI) in the presence of Cr(III) [Meija 2006a] or MeHg+ in the presence of Hg(II) [Hintelmann 1997]. To address these challenges and obtain unbiased estimates of Cr(VI) or MeHg+ concentration, the basic equations of isotope dilution have to be adjusted to correct for the possible analyte formation [Meija 2008a]. Several mathematical strategies now exist to address the analyte formation and degradation using isotope dilution. Recently Rodríguez-González et al. compared the numerical performance of the four existing approaches for multiple spiking species-specific isotope dilution analysis using butyltin determination in sediments as an example [Rodríguez-González 2007]. While all of these strategies have been shown to give identical numerical results for the initial amount of substances in the sample, the coefficients that describe the inter-conversion differ. Such differences are solely due to the unrealized inconsistencies in current isotope dilution equations, which are discussed below.
To describe species transformation during the analysis, many analytical chemists have long ailed—what matters is what something is, not what it is called [Dumon 1993]. As a result, to describe the formation of CH3Hg+ from Hg(II) there are a gamut of vague terms, such as “specific methylation” [Hintelmann 1997], “accidental formation rate” [Hintelmann 1999], “specific rate of methylation” [Hintelmann 1995], “degree of methylation” [Qvarnström 2002], “methylation yield” [Point 2007], “methylation rate” [Lambertsson 2001] and “methylation activity” [Eckley 2006], just to name few. As an example, one can find four different synonyms (methylation factor, yield, rate and intensity) for a single dimensionless variable used to quantify the methylation of Hg(II) in a recent report [Point 2008]. One cannot but wonder about the precise meaning of these variables.
The variables that quantify the analyte formation are increasingly used by chemists to evaluate analytical protocols. As a result, species inter-conversion coefficients have been used in recent years along with the degradation-corrected amount of analytes. For example, U.S. Environmental Protection Agency has recommended that isotope dilution results be discarded when the values of the inter-conversion coefficients exceed certain threshold [USEPA 1998]. Further to the frivolous naming conventions, it turns out that definitions of these coefficients remain murky at best despite the volume of recent studies that rest on the numerical values aimed at quantification of the analyte inter-conversion [Point 2007; Point 2008; Monperrus 2008].
In order to fully grasp the intricacies of the isotope dilution for inter-converting species, the basic building principle of isotope dilution equations are reviewed herein. For a closed two component system, the amount balance of both analytes before (nA,B0) and after (nA,B) the conversion can be generalized in the form of the following two expressions using amount transfer coefficients, ki:nA=nA0·k1+nB0·k2  [1]nB=nA0·k3+nB0·k4  [2]As an example, equations developed by Kingston et al. [Kingston 1998] (and Meija et al. [Meija 2006a]) for the inter-conversion of two species take the following form:nA≡nA0·(1−α1)+nB0·α2  [3]nB≡nA0·α1+nB0·(1−α2)  [4]Regardless of the model used to describe the inter-conversion, the resulting equations must obey one of the most fundamental laws of nature—conservation of the amount:nA+nB=nA0+nB0  [5]However, the conservation of the amount seems to be often neglected in isotope dilution equations. Qvarnström and Frech, for example, attain the following expressions for the Hg(II)/CH3Hg+ system [Qvarnström 2002]:nHg(II)0≡nHg(II)−nMeHg+·b2  [6]nMeHg+0≡nMeHg+−nHg(II)·b1  [7]
The above equations violate the amount balance of Hg(II) and CH3Hg+, i.e. does not lead to Eq. [5]. Only if b1=b2=0 does the above equation fulfill the conservation of amount. Numerically these coefficients (b1, b2) are identical to the “degradation factors”, Fi, of Rodríguez-González et al. [Rodríguez-González 2007; Rodríguez-González 2004]. For a two-component system consider the following amount balance equations:nA≡nA0·(1−F1)+nB0·F2(1−F1)  [8]nB≡nA0·F1(1−F2)+nB0·(1−F2)  [9]Violation of amount balance in this system is also evident as the sum of these two equations does not lead to Eq. [5]. Due to error cancellation, the values for the initial amount of analytes (n0) are unbiased even though the underlying amount balance models are incorrect in most of these cases. Violation of amount balance leads to incorrect estimates of the amount of analytes present in solution at the time of analysis (nA,B). An in silico experiment that illustrates this corollary is shown in Table 1.
TABLE 1Amount of Hg(II) and CH3Hg+ from a sample initially containing1.0 mol of each compound*Isotope dilutionConversionmodelcoefficientsEquationsn[Hg(II)]n[CH3Hg]Hintelmann et al.b1, 2 = 0.500, 0.667[6], [7]2.50 mol2.25 molRodríguez-F1, 2 = 0.500, 0.667[8], [9]0.83 mol0.50 molGonzález et al.Kingston et al.α1, 2 = 0.250, 0.500[3], [4]1.25 mol0.75 molMeija et al.α1, 2 = 0.250, 0.500[3], [4]1.25 mol0.75 mol*Consider 1.0 mol of 201Hg(II) that is mixed with 1.0 mol of CH3198Hg+. Then, 50% of Hg(II) is transformed into CH3Hg+ resulting in 0.5 mol 201Hg(II), 0.5 mol CH3201Hg+ and 1.0 mol CH3198Hg+. Then, 50% of the CH3Hg+ is converted into Hg(II) yielding to the following: 0.50 mol CH3198Hg+, 0.25 mol CH3201Hg+, 0.50 mol 198Hg(II) and 0.75 mol 201Hg(II). Amount of Hg(II) and CH3198Hg+ at this point is 1.25 mol and 0.75 mol respectively. Using these “observed” isotope patterns of Hg(II) and CH3Hg+, any of the four existing isotope dilution models can now be used to calculate the inter-conversion coefficients and the amount of these compounds after inter-conversion (as per Eqs. [3], [4] or [6], [7] or [8], [9]).
As a result of amount imbalance (Eqs. [8], [9]) the coefficients Fi and αi are different (see Table 1). Analytical relationship between these is as follows:
                              F          1                =                                                            α                1                                            1                -                                  α                  2                                                      ⁢                                                  ⁢            and            ⁢                                                  ⁢                          F              2                                =                                    α              2                                      1              -                              α                1                                                                        [        10        ]            From here the numerical discrepancy between F1 and α1 or F2 and α2, as recently noted by Rodríguez-González et al. [Rodríguez-González 2007] (and later dismissed [Point 2008]), is evident. When all αi are large, the numerical difference between both notations becomes obvious [Meija 2006a]. Conceptually, the coefficients α1 and α2 consistently describe the final state of inter-converting species whereas the coefficients of Hintelmann et al. and Rodríguez-González et al. link the degradation non-corrected (i.e. wrong) amount of species to the correct ones. Clearly, the latter coefficients have no meaning apart from the role as numerical correction factors.
While the above caveats do not diminish the capability of multiple spiking isotope dilution methods to infer about the species inter-conversion, it clearly shows that fundamental definitions and notation is urgently needed.
One Isotope Pattern, Several Explanations
Central to the isotope dilution paradigms is the idea that each measured isotope pattern determines a unique set of analyte concentrations [Meija 2008a]. While it is true, the same cannot be said about the analyte inter-conversion coefficients. Consider the inter-converting system of species A and B with their initial amounts of 5 mol and 1 mol respectively. Isotope patterns of these species are χA,0=(1.000, 0.000) and χB,0=(0.000, 1.000). These two compounds were mixed together and, after certain inter-conversion process, the isotope patterns of both of these compounds was χA=(0.882, 0.118) and χB=(0.714, 0.286).
Inter-conversion reactions can occur via different routes. For example, the reactions A→B and B→A can occur sequentially or simultaneously. In the case of Hg(II) and CH3Hg+, methylation of Hg(II) can occur prior to demethylation or vice versa. Both of these reactions can also occur simultaneously. All three scenarios, if applied to the observed isotope patterns, lead to drastically different explanations of the inter-conversion process. The above system, for example, can be explained with the gamut of values for the fraction of B that has converted into A and vice versa depending on the nature of the inter-conversion (FIG. 1). It is clear that the answer to the question what is the fraction of compound A that converts into B can be obtained only if the mechanism of the inter-conversion is known. This, however, is often not the case for systems where double-spiking isotope dilution is currently used in practice.
Extent of Conversion, ξ
The central aim of quantifying the inter-conversion of species is the measurement of the total amount of a compound that has converted into another species. This relates to the formal IUPAC definition of the extent of conversion (or reaction), ξ, as the number of chemical transformations divided by the Avogadro constant [IUPAC Compendium; Laidler 1996]. This is essentially the amount of chemical transformations. If a single forward reaction ν1Hg(II)→ν2MeHg+ occurs in a closed system and has known time-independent stoichiometry, the extent of conversion at any given time (t) is defined by the following particular expression:
                              ξ                      Hg            ->            MeHg                          =                                            n                              Hg                ->                MeHg                                                    v              1                                =                                                    n                                  Hg                  ⁡                                      (                    II                    )                                                  0                            -                              n                                  Hg                  ⁡                                      (                    II                    )                                                  t                                                    v              1                                                          [        11        ]            
Extent of conversion quantifies the amount of Hg(II) methylated to CH3Hg+ and, by definition, depends on the mechanism of the inter-conversion. Rather overlooked is the interpretation of the extent of reaction for reversible reactions since Eq. [11] no longer applies. For reversible process, such as Hg(II)⇄CH3Hg+, the total amount of Hg(II) that has been methylated to CH3Hg+, i.e. ξ of the forward reaction, is also a function of the forward and backward rate constants k1 and k2:
                                          ξ            ->                    =                                    n                              Hg                ->                MeHg                                      =                                          ∫                t                            ⁢                                                k                  1                                ⁢                                                      n                                          Hg                      ⁡                                              (                        II                        )                                                                              ⁡                                      (                    t                    )                                                  ⁢                                  ⅆ                  t                                                                    ⁢                                  ⁢                              ξ            ←                    =                                    n                              MeHg                ->                Hg                                      =                                          ∫                t                            ⁢                                                k                  2                                ⁢                                                      n                    MeHg                                    ⁡                                      (                    t                    )                                                  ⁢                                  ⅆ                  t                                                                                        [        12        ]            Integrating these expressions leads to the following:
                                          ξ            ->                    =                                    n                              Hg                ->                MeHg                                      =                                                            k                  1                                                                      k                    1                                    +                                      k                    2                                                              ⁡                              [                                                                            n                                              Hg                        ⁡                                                  (                          II                          )                                                                    0                                        ⁡                                          (                                              1                        +                                                                              k                            2                                                    ⁢                          t                                                                    )                                                        +                                                            n                      MeHg                      0                                        ⁢                                          k                      2                                        ⁢                    t                                    -                                                            n                                              Hg                        ⁡                                                  (                          II                          )                                                                                      ⁡                                          (                      t                      )                                                                      ]                                                    ⁢                                  ⁢                              ξ            ←                    =                                    n                              MeHg                ->                Hg                                      =                                                            k                  2                                                                      k                    1                                    +                                      k                    2                                                              ⁡                              [                                                                            n                      MeHg                      0                                        ⁡                                          (                                              1                        +                                                                              k                            1                                                    ⁢                          t                                                                    )                                                        +                                                            n                                              Hg                        ⁡                                                  (                          II                          )                                                                    0                                        ⁢                                          k                      1                                        ⁢                    t                                    -                                                            n                      MeHg                                        ⁡                                          (                      t                      )                                                                      ]                                                                        [        13        ]            We also introduce the relative extent of conversion, ξr,A→B, as the amount of A that converts into B during the course of reaction relative to the initial amount of A:
                              ξ                      r            ,                          A              ->              B                                      =                              ξ                          A              ->              B                                            n            A            0                                              [        14        ]            The concept of reaction extent is a ramification of chemical kinetics and is usually not used in practice of analytical chemistry in simultaneous inter-conversion processes. Rather, the mere difference between the initial and measured amounts (at time t) is commonly used as a substitute for the total amount of A that has converted into B. As an example, the fate of methylmercury in biota is often elucidated from inter-conversion coefficients (Hintelmann [1997; Hintelmann 1995] presumed to represent the total amount of Hg(II) methylated and CH3Hg+ demethylated, i.e. extent of (de)methylation. It is important to dissociate the extent of conversion with any of the inter-conversion factors stemming from the isotope dilution results. Traditionally the extent of conversion has been associated with the numerical values of the correction factors [Rodriguex-Gonzalez 2007]. While the definition of the extent of conversion can be realized in practice, the underlying mechanism of the inter-conversion must be specified. In certain cases it is possible to deduce an educated guess regarding this. For example, Cr(VI) is stable in alkaline medium and yeast digestion at 95° C. for the analysis of Cr(III) and Cr(VI) suggests that the oxidation of Cr(III), if any, will occur before the reduction of Cr(VI) once the digests are neutralized. In other cases, such as CH3Hg+/Hg(II), the inter-conversion mechanisms are more complex and currently not well understood.Degree of Conversion, α
Degree of conversion is often used to describe bi-directional processes such as ionization of electrolytes or dissociation of acids. In accord with the existing chemical nomenclature, degree of conversion of compound A (αA,B) is the amount fraction of A present in its converted form B [IUPAC Compendium]. In Hg(II)⇄CH3Hg+ system, for example, degree of methylation is the amount of Hg(II) present as CH3Hg+ divided to the initial amount of Hg(II).
Notation of Species Inter-Conversion
In isotope dilution, the inter-conversion of analytes can be modeled via two conceptually different approaches: using macroscopic and microscopic degrees of reactions (thermodynamic approach) and rate constants (kinetic approach) [Boyd 1977]. In the thermodynamic approach the amount balance of the involved compounds is established by comparing the isotope patterns of the involved species before and after the potential inter-conversion using degree of reaction (conversion). The kinetic approach, however, describes the analyte formation and loss using explicit assumptions as to how the inter-conversion occurs in time, i.e. simultaneously or sequentially, involving first or other order kinetics. Both of these approaches exist in the literature. Within these approaches, the analyte inter-conversion is described using “amount fraction of species that converts into another species” [Rahman 2004] and “amount fraction of species that [has] converted into another species” [Rodríguez-González 2004; Rodríguez-González 2005a; Rodríguez-González 2007].
Phenomenological (Macroscopic) Notation
The thermodynamic approach to species inter-conversion describes the inter-conversion using phenomenological degree of conversion. In a two-component system we denote these coefficients as α1 and α2. For example, α1=0.20 means that 20% from the initial amount of compound A exists as B at the time of analysis given that the system (A, B) is closed. This, however, does not necessarily mean that 20% of compound A has converted into B. Hence the distinction between the degree of conversion (fraction of species that exists in the form of another species) and relative extent of conversion (fraction of species that has converted into another species). The amount balance of substances A and B before and after their inter-conversion can be written using degree of conversion, as in Eqs. [3] and [4], where α1 and α2 merely account for the difference between the initial and final amount of both species. As such, the phenomenological degrees of reaction can be obtained for every system, regardless the mechanism of the inter-conversion. Isotope dilution models developed by Kingston et al. [Kingston 1998] follow this notation and so does the matrix approach of Meija et al. [Meija 2006a]. We note that the traditional interpretation of α1 and α2 as “the fraction of Cr(III) that converts to Cr(VI) and vice versa” [Rahman 2004] or “the percentage of Cr(III) oxidized to Cr(VI) and vice versa” [USEPA 1998; USEPA 2007] is false. It must be replaced with “the fraction of the initial amount of Cr(III) that is Cr(VI) at the time of analysis and vice versa” [Jereb 2003]. It is important to stress that the phenomenological degrees of conversion will sustain their meaning only when the system of inter-converting species is known to be closed. However, amount balance experiments in this area are performed seldom.
Microscopic Notation
Microscopic approach to amount balance proceeds by knowing/assuming the mechanism of the inter-conversion. There are various ways two compounds may convert into each other as shown in FIG. 2. Consider the system where reactions A→B and B→A occur at different time periods (in that order) as in Scheme 2.3 of FIG. 2. Using the microscopic degree of reactions (αm1, αm2), the amount balance of the involved species before (n0) and after (n1) the first reaction step for this system can be written as follows:nA1=nA0·(1−αm1)  [15]nB1=nB0+nA0·αm1  [16]After the second reaction step, however, the amount of A and B are as follows:nA≡nA1+nB1·αm2=nA0·(1−αm1αm2−αm1)+nB0·αm2  [17]nB≡nB1·(1−αm2)=nA0·αm2(1−αm1)+nB0·(1−αm2)  [18]In other words, the microscopic degrees of reaction are the answer to a hypothetical question “how much of both species have converted into one another at each step of the conversion process”. The relationship between the phenomenological (thermodynamic) and microscopic (kinetic) degrees of reaction depends on the conversion mechanism and for the above example system (Scheme 2.3 of FIG. 2) it is the following:
                              α                      m            ⁢                                                  ⁢            1                          =                                                            α                1                                            1                -                                  α                  2                                                      ⁢                                                  ⁢            and            ⁢                                                  ⁢                          α                              m                ⁢                                                                  ⁢                2                                              =                      α            2                                              [        19        ]            One of the main pitfalls of the microscopic notation is the implicit idea that the species inter-conversion can be described using the constant degrees of reaction whereas the degree of reaction is not a constant over the course of any chemical reaction, regardless of their kinetic order (see Eq. [22] for example). Thus, in the context of amount balance equations in isotope dilution, it is only meaningful to use the phenomenological and not microscopic degree of reaction as species inter-conversion constants in Eqs. [1]-[2].Kinetic Notation
Consider two analytes that can simultaneously inter-convert into each other according to first-order reactions A⇄B with rate constants kA→B and kB→A. We denote these as kA,B and kB,A. For such system, changes in the amount of these compounds can be established by the use of two coupled ordinary differential equations in accord to the law of ‘active masses’:
                    {                                                                                                  ⅆ                                          n                      A                                                                            ⅆ                    t                                                  =                                                                            -                                              k                                                  A                          ,                          B                                                                                      ⁢                                                                  n                        A                                            ⁡                                              (                        t                        )                                                                              +                                                            k                                              B                        ,                        A                                                              ⁢                                                                  n                        B                                            ⁡                                              (                        t                        )                                                                                                                                                                                                          ⅆ                                          n                      B                                                                            ⅆ                    t                                                  =                                                                            +                                              k                                                  A                          ,                          B                                                                                      ⁢                                                                  n                        A                                            ⁡                                              (                        t                        )                                                                              -                                                            k                                              B                        ,                        A                                                              ⁢                                                                  n                        B                                            ⁡                                              (                        t                        )                                                                                                                                                    [        20        ]            This system can be solved using the eigenvalue/eigenvector method [Blanchard 2006]. At time t we observe the following amount of A and B:
                    {                                                                              n                  A                                =                                                                                                                              k                                                      B                            ,                            A                                                                          +                                                                              k                                                          A                              ,                              B                                                                                ⁢                                                      ⅇ                                                                                          -                                                                  k                                  Σ                                                                                            ·                              l                                                                                                                                                  k                        Σ                                                              ⁢                                          n                      A                      0                                                        +                                                                                                              k                                                      B                            ,                            A                                                                          -                                                                              k                                                          B                              ,                              A                                                                                ⁢                                                      ⅇ                                                                                          -                                                                  k                                  Σ                                                                                            ·                              l                                                                                                                                                  k                        Σ                                                              ⁢                                          n                      B                      0                                                                                                                                                                n                  B                                =                                                                                                                              k                                                      A                            ,                            B                                                                          -                                                                              k                                                          A                              ,                              B                                                                                ⁢                                                      ⅇ                                                                                          -                                                                  k                                  Σ                                                                                            ·                              l                                                                                                                                                  k                        Σ                                                              ⁢                                          n                      A                      0                                                        +                                                                                                              k                                                      A                            ,                            B                                                                          +                                                                              k                                                          B                              ,                              A                                                                                ⁢                                                      ⅇ                                                                                          -                                                                  k                                  Σ                                                                                            ·                              l                                                                                                                                                  k                        Σ                                                              ⁢                                          n                      B                      0                                                                                                                              [        21        ]            where kΣ=kA,B+kB,A. The (simplified) reversible reaction model has been applied before to obtain the rate constants of Hg(II) methylation and CH3Hg+ demethylation reactions [Rodriguez Martin-Doimeadios 2004]. Comparison of the obtained expression with Eqs. [3]-[4] leads to the following relationship between the phenomenological degrees of conversion and the rate constants for the simultaneous process:
                              α                      A            ,            B                          =                                                            k                                  A                  ,                  B                                                            k                Σ                                      ⁢                          (                              1                -                                  ⅇ                                                            -                                              k                        Σ                                                              ·                    l                                                              )                        ⁢                                                  ⁢            and            ⁢                                                  ⁢                          α                              B                ,                A                                              =                                                    k                                  B                  ,                  A                                                            k                Σ                                      ⁢                          (                              1                -                                  ⅇ                                                            -                                              k                        Σ                                                              ·                    l                                                              )                                                          [        22        ]            Values of α1 and α2 can be obtained experimentally from the phenomenological isotope dilution models, hence, the rate constants can be calculated from thereof:
                                          k                          A              ,              B                                ·          t                =                                            -                              α                                  A                  ,                  B                                                                                    α                                  A                  ,                  B                                            +                              α                                  B                  ,                  A                                                              ⁢                      ln            ⁡                          (                              1                -                                  α                                      A                    ,                    B                                                  -                                  α                                      B                    ,                    A                                                              )                                                          [        23        ]                                                      k                          B              ,              A                                ·          t                =                                            -                              α                                  B                  ,                  A                                                                                    α                                  A                  ,                  B                                            +                              α                                  B                  ,                  A                                                              ⁢                      ln            ⁡                          (                              1                -                                  α                                      A                    ,                    B                                                  -                                  α                                      B                    ,                    A                                                              )                                                          [        24        ]            If αA,B+αB,A<<1, kA,B·t≈αA,B and kB,A·t≈αB,A since ln x≈(x−1) when x≈1. Solving the integral for the relative extent of conversion (noting that the constant of integration is not zero) leads to expressions that can be expressed using degrees of the individual conversions and the initial amount of both substances:
                              ξ                      r            ,                          A              ->              B                                      =                                                            α                1                            ⁢                              α                2                                                                    (                                                      α                    1                                    +                                      α                    2                                                  )                            2                                ⁡                      [                                                            (                                                            α                      1                                        +                                          α                      2                                                        )                                ⁢                                  (                                                                                    α                        1                                                                    α                        2                                                              -                                                                  n                        B                        0                                                                    n                        A                        0                                                                              )                                            -                                                (                                                                                    n                        B                        0                                                                    n                        A                        0                                                              +                    1                                    )                                ⁢                                  ln                  ⁡                                      (                                          1                      -                                              α                        1                                            -                                              α                        2                                                              )                                                                        ]                                              [        25        ]                                          ξ                      r            ,                          B              ->              A                                      =                                                            α                1                            ⁢                              α                2                                                                    (                                                      α                    1                                    +                                      α                    2                                                  )                            2                                ⁡                      [                                                            (                                                            α                      1                                        +                                          α                      2                                                        )                                ⁢                                  (                                                                                    α                        2                                                                    α                        1                                                              -                                                                  n                        A                        0                                                                    n                        B                        0                                                                              )                                            -                                                (                                                                                    n                        A                        0                                                                    n                        B                        0                                                              +                    1                                    )                                ⁢                                  ln                  ⁡                                      (                                          1                      -                                              α                        1                                            -                                              α                        2                                                              )                                                                        ]                                              [        26        ]            When α1+α2<<1, relative extent of conversion is approximately equal to the degree of conversion, i.e. ξr,A→B≈α1 and ξr,B→A≈α2.Numerical Example
The extent of conversion, i.e. the amount of compound that has been transformed into another, can be obtained by multiplying relative extent of conversion with the initial amount of the analyte. Consider an in silico experiment where 5 mol of 201Hg(II) and 0.01 mol of CH3198Hg+ are added to a mercury-free solution of organic matter. After 7 hours of simultaneous first-order reactions, Hg(II)⇄CH3Hg+, the isotope patterns (x198, x201) of) both compounds was measured to be xHg=(0.00101, 0.99899) and xMeHg=(0.16564, 0.83436). Results calculated from these observations are summarized in Table 2.
In this example, degree of CH3Hg+ demethylation is 50% whereas the relative amount of CH3Hg+ demethylated (ξr,←) is by far larger, i.e. 150%. Hence, the amount of CH3Hg+ demethylated is underestimated by a factor of three. Furthermore, the ratio of the methylation/demethylation extent, ξ→/ξ←=2.35, is significantly different from the conventional methylation-to-demethylation ratio M/D=10.0 [Hintelmann 1997; Qvarnström 2002; Monperrus 2007], which is equal to (b1n1*)/(b2n2*) or (F1n1*)/(F2n2*).
TABLE 2Quantitation of Hg(II)/CH3Hg+ inter-conversion*QuantityValueEquationDegree of methylationα1 = 0.005019[3]-[4]and demethylation**α2 = 0.5019Amount of Hg(II) andn(Hg) = 4.9799 mol[3]-[4]CH3Hg+ after 7 hn(CH3Hg+) = 0.0301 molMethylation and demethylationk1 = 0.0010 h−1[23]-[24]rate constantsK2 = 0.1000 h−1Relative extent of methylationξr,→ = 0.00698[25]-[26]and demethylationξr,← = 1.485Extent of methylationξ→ = 0.0349 mol[14]and demethylationξ← = 0.0148 mol*Hg(II)/CH3Hg+ inter-conversion has been modeled in silico by solving Eq. [21] with rate constants k1 = 0.0010 h−1 and k2 = 0.1000 h−1. Amounts of both analytes and the rate constants roughly mimic the conditions of typical estuarine waters.**Obtained using the double spiking isotope dilution calculations [Meija 2006a].
While isotope dilution has been successfully used to estimate amount of species corrected for the analyte degradation and formation during the analysis, prior art underlying mathematical models have not been scrutinized. As a result, proper interpretation and clear definitions of the inter-conversion coefficients has been overlooked despite the recent widespread use of these coefficients in analytical method development. We recommend the use of the species inter-conversion coefficients consistent with the current IUPAC guidelines as summarized in Table 3, which will be used throughout the present specification. Surprisingly, the same applies to the amount of analyte at the time of analysis. The consequence of the above exposition is that the extent of the species inter-conversion can only be quantified when its mechanism is known. Parallels of this truism are found in quantitative analysis—it is only possible to quantify a compound whose identity is known, i.e. “quantification of an unknown compound” is an absurd (albeit often used) phrase [Meija 2008a].
TABLE 3Quantities to describe chemical transformationsNameSymbolDefinitionSI unitExtent of reaction1, 2ξA→BNumber of chemicalmoltransformation ν1A→ν2B dividedby the Avogadro constantRelative extentξr, A→BExtent of reaction ν1A→ν2B divided1of reactionby the initial amount of ADegree of reaction3, 4αA→BAmount fraction of A present in its1converted form BCorrection factor5FNumerical factor by which the1uncorrected result of ameasurement is multiplied tocompensate for systematic error1Equation ξA = (nA − nAo)/νA applies only to a single reaction, νAA→νBB, occurring in a closed system. Here nAo is the initial amount of the entity A, nA is its amount at time t, and νA is the stoichiometric number for that entity in the reaction equation as written [IUPAC Compendium].2Extent of reaction is often confused with the degree of reaction.3Most common interpretations of this variable are degree of dissociation, ionization and polymerization.4When the term “reaction” covers multitude of chemical reactions, α represents phenomenological (macroscopic) degree of reaction. To distinguish between the microscopic and macroscopic degrees of reaction, subscript “m” can be added to denote the former.5Uncorrected result refers to the result that is obtained using isotope dilution equations that ignore any analyte formation. Systematic error here refers only to the error introduced by neglecting the analyte formation [International Organization for Standardization 1993].Uncertainties
Inter-conversion of analytes is inevitably accompanied with the loss of information) that can be extracted from the isotope patterns. Therefore, any corrections for analyte inter-conversion are performed at the expense of the precision of the obtained amount of the inter-converting analytes. Consequently, there is a natural, predictable limit to the applicability of multiple-spiking isotope dilution methods.
As the importance of analyte inter-conversions was established and multiple spiking isotope dilution was employed to correct for the inter-conversion [Point 2007; Monperrus 2008; Kingston 1998] little attention has been devoted regarding the fundamental limitations and consequences of such corrections. For example, how does the inter-conversion affect the uncertainty of the analytical results and what role does the amount ratio of the inter-converting species play? While intuitively it has been known that inter-conversion degrades the precision of the amount estimates [USEPA 1998] mathematical analysis of this phenomenon is clearly lacking [Monperrus 2008], given the fact that the fundamental aspects of multiple-spiking isotope dilution are not well understood in the first place as discussed above.
There remains a need in the art for a method multiple spiking isotope dilution analysis for mass spectrometry that provides precise and simultaneous characterization of substances in a sample, and particularly a method in which uncertainties in the characterization can be accurately estimated.